Schauder Estimate for Quasilinear Discrete PDEs of Parabolic Type

We investigate quasilinear discrete PDEs ∂ t u = Δ N φ ( u ) + K f ( u ) ∂ _t u = Δ ^N φ (u)+ Kf(u) of reaction-diffusion type with nonlinear diffusion term defined on an n n -dimensional unit torus discretized with mesh size 1 N 1N for N ∈ N N∈ {N} , where Δ N Δ ^N is the normalized discrete Laplacian, φ φ is a strictly increasing C 5 C^5 function and f f is a C 1 C^1 function. We establish L ∞ L^∞ bounds and space-time Hölder estimates, both uniform in N N , of the first and second spatial discrete derivatives of the solutions. In the equation, K > 0 K>0 is a large constant and we show how these estimates depend on K K